Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1358 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #331 |
| commutator | C42 | C2×C4×C11 |
| Frattini, cosocle | C4×C8 | C22×C11 |
| prime radical | C22×C11 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,20,16,19,28,35,17,22,29,38,31,21,39,37)(14,23,24,40,25,34,41,43,26,33,42,36,30,18,27,32);(12,14,17,26)(13,18,22,34)(15,27,29,41)(16,30,31,25)(19,33,21,23)(20,36,38,40)(24,28,42,39)(32,35,43,37)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 90);